For some preliminary results on the Lebesgue and Sobolev spaces with variable exponent, we only give the definition of \(L^{p(.)}(\Omega )\) , for more details, see [1]}, [2]} or monographs [3]}, [4]}. For an open \(\Omega \subset \mathbb {R}^N\) , let \(p:\Omega \rightarrow [1,+\infty )\)  be a measurable function such that
\(1<p^-=\mathrm {ess\,inf}\,p,\quad p^{+}=\mathrm {ess\,sup}\,p<+\infty .\) 
